×

Duality for projective curves. (English) Zbl 0766.14021

Let \(X\) be a non-degenerate curve in \(\mathbb{P}^ n\) (= projective \(n\)- space) over an algebraically closed field. Let \(b_ 0<b_ 1<\cdots<b_ n\) be the order sequence of \(X\), consisting of the possible intersection multiplicities of \(X\) with hyperplanes at a general point of \(X\). Let \((\mathbb{P}^ n)'\) be the dual projective space and \(X'\subset(\mathbb{P}^ n)'\) be the dual curve, parametrizing the osculating hyperplanes of \(X\). Let \(\gamma:X\to X'\) be the Gauss map. The authors determine, for each \(j=0,\dots,n-1\), an integer \(s_ j\) (depending on the order sequences of \(X\) and \(X')\) such that, for the general point \(P\) of \(X\), the osculating \(s_ j\)-plane of \(X'\) at \(\gamma(P)\) is contained in the dual of the osculating \(j\)-plane of \(X\) at \(P\). Moreover, under some divisibility assymptions, the osculating \((s_ j+1)\)-plane of \(X'\) at \(\gamma(P)\) is not contained in the dual of the osculating \(j\)-plane of \(X\) at \(P\). As a consequence, a characterisation of the non-reflexive curves that coincide with their bidual is obtained. For example, if \(X\) is a nonreflexive smooth plane curve of degree at least 4, then \(X''=X\) if and only if \(X\) is defined over a finite field \(\mathbb{F}_ q\) and, denoting by \(F\) the \(\mathbb{F}_ q\)-Frobenius map, \(F(P)\) belongs to the tangent line to \(X\), at \(P\), for every point \(P\) of \(X\).

MSC:

14H50 Plane and space curves
14N05 Projective techniques in algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arbarello, E. et al., ?Geometry of Algebraic Curves,? vol. I. Springer, New York (1985). · Zbl 0559.14017
[2] Bertini, E., ?Introduzione alla geometria proiettiva degli iperspazi.? Principato, Messina, (1923). · JFM 49.0484.08
[3] Garcia, A.,The curves y n=f(x) over finite fields. Arch. Math.54 (1990), 36-44. · Zbl 0687.14018 · doi:10.1007/BF01190666
[4] Garcia, A., Voloch, J.F.,Fermat Curves over Finite Fields, J. Number Theory30 (1988), 345-356. · Zbl 0671.14012 · doi:10.1016/0022-314X(88)90007-8
[5] Hartshorne, R., ?Algebraic Geometry,? Springer, New York. (1977). · Zbl 0367.14001
[6] Hasse, H., Schmidt, F.K.,Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten, J. reine angew. Math.177 (1937), 215-237. · Zbl 0017.10101
[7] Hefez, A.,Non-reflexive curves, Compositio Math.69 (1989), 3-35. · Zbl 0706.14024
[8] Hefez, A., Kakuta, N.. To appear.
[9] Hefez, A., Voloch, J.F.,Frobenius non-classical curves, Arch. Math.54 (1990), 263-273. · Zbl 0662.14016 · doi:10.1007/BF01188523
[10] Kaji, H.,On the Gauss maps of space curves in characteristic p, Compositio Math.70 (1989), 177-197. · Zbl 0692.14015
[11] Kleiman, S.L.,Tangency and duality, Conf. Proc., Canadian Math. Soc.6 (1986), 163-225. · Zbl 0601.14046
[12] Laksov, D.,Wronskians and Plücker formulas for linear systems on curves, Ann. Sci. École Norm. Sup.17, (1984), 45-66. · Zbl 0555.14008
[13] Piene, R.,Numerical characters of a curve in projective n-space, Real and Complex Singularities, Oslo, Sijthoff and Nordhoff. · Zbl 0375.14017
[14] Schmidt, F.K.,Die Wronskische Determinante in beliebigen differenzierbaren Funktionenkörpern, Math. Z.45 (1939), 62-74. · Zbl 0020.10201 · doi:10.1007/BF01580273
[15] Schmidt, F.K.,Zur arithmetischen Theorie der algebraischen Funktionen II. Allgemeine Theorie der Weierstrasspunkte, Math. Z.45 (1939), 75-96. · Zbl 0020.10202 · doi:10.1007/BF01580274
[16] Severi, F., ?Trattato di Geometria Algebrica.? Zanichelli, Bologna, (1926). · JFM 52.0650.01
[17] Stöhr, K.O., Voloch, J.F.,Weierstrass points and curves over finite fields, Proc. London Math. Soc.52 (1986), 1-19. · Zbl 0593.14020 · doi:10.1112/plms/s3-52.1.1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.